Electronic Journal of Qualitative Theory of Differential Equations2018, No. 68, 1–21; https://doi.org/10.14232/ejqtde.2018.1.68 www.math.u-szeged.hu/ejqtde/

Multiple positive solutions for Schrödinger problemswith concave and convex nonlinearities

Xiaofei CaoB 1 and Junxiang Xu2

1Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huaian, 223003, China2Faculty of Mathematics, Southeast University, Nanjing 210096, China

Received 19 November 2017, appeared 5 August 2018

Communicated by Petru Jebelean

Abstract. In this paper, we consider the multiplicity of positive solutions for a classof Schrödinger equations involving concave-convex nonlinearities in the whole space.With the help of the Nehari manifold, Ekeland variational principle and the theory ofLagrange multipliers, we prove that the Schrödinger equation has at least two positivesolutions, one of which is a positive ground state solution.

Keywords: Schrödinger problem, Nehari manifold, Ekeland variational principle.

2010 Mathematics Subject Classification: 35A01, 35A15.

1 Introduction and main results

This paper concerns the multiplicity of positive solutions for the following Schrödinger equa-tion

−4u + V(x)u = f (x)|u|q−2u + g(x)|u|p−2u in RN ,

u ∈ H1 (RN) ,(1.1)

where 1 < q < 2 < p < 2∗ (2∗ = ∞ if N = 1, 2 and 2∗ = 2N/(N − 2) if N ≥ 3) andV(x), f (x), g(x) satisfy suitable conditions.

There are many works on nonlinearity of concave-convex type under various conditionson potential V(x). When V(x) ≡ 0, Equation (1.1) is considered in a bounded domain. Thisproblem can date back to the famous work of Ambrosetti–Brezis–Cerami in [1], where theauthors considered the following problem

−4u = λ|u|q−2u + |u|p−2u in Ω,

u ∈ H10(Ω),

(1.2)

where Ω ⊂ RN is a bounded domain, 1 < q < 2 < p ≤ 2∗. They proved that Equation(1.2) has at least two positive solutions for suffciently small λ > 0. In this case, the compactembedding H1

0(Ω) → Lp(Ω) (p ∈ [2, 2∗)) plays an important role; for more general results

BCorresponding author. Email: caoxiaofei258@126.com

2 X. Cao and J. Xu

in bounded domains see [4, 5, 8, 13, 18, 24, 27] and their references. In the whole space RN

some authors concerned Equation (1.1) with V(x) satisfying suitable conditions such that theembedding

X :=

u ∈ H1(

RN)

:∫

RNV(x)|u(x)|2dx < +∞

→ Lp

(RN)

, p ∈ [2, 2∗) , (1.3)

is compact. For example, Bartsch and Wang [2] first introduced the following weaker condition

(V) V(x) ∈ C(RN , R

), V0 := infRN V(x) > 0 and for any M > 0, there exists a constant

r0 > 0 such that meas (x ∈ Br0(y) : V(x) ≤ M) → 0 as |y| → +∞, where Br0(y) de-notes the ball centered at y with radius r0 and mea s the Lebesgue measure in RN .

For some results in this area, we also refer to [14, 21].If the potential function V(x) is bounded, the embedding (1.3) is not compact; in the case

of the constant potential, i.e., V(x) is a positive constant in Equation (1.1), we can refer to[25, 26, 28]. However, we do not know any results for Equation (1.1) with both V(x) and g(x)bounded functions. A direct extension to the case V(x) and g(x) bounded functions is facedwith difficulties. On the one hand, because the nonlinearity is a combination of the concaveand convex terms, estimating the critical value by suitable autonomous equation becomescomplex. On the other hand, since both V(x) and g(x) are bounded functions, the proof ofthe (PS) condition satisfied for the critical value in suitable range becomes delicate. In thispaper, we are concerned about Equation (1.1) with both V(x) and g(x) bounded functionson the basis of variational arguments. If V(x), f (x) and g(x) satisfy the suitable conditions,we prove multiple positive solutions for equation (1.1) under the quantitative assumption.Up to now, there is a lot of papers considered different problems and obtained the relevantresults under the quantitative assumption, see [6, 7, 12, 29] for Kirchhoff problems, [15, 26, 27]for Schrödinger problems and [16] for Schrödinger–Maxwell problems. For example, Wu [27]considered the following Schrödinger problem:

−4u = f (x)|u|q−2u + (1− g(x))|u|2∗−2u in Ω,

u = 0, in ∂Ω,

where 1 < q < 2, 2∗ = 2N/(N − 2)(N ≥ 3), Ω ⊂ RN is a bounded domain with smoothboundary and the weight functions f , g ∈ C(Ω) satisfy the suitable conditions. Then thereexists λ0 > 0 such that if ‖ f+‖Lq∗ < λ0, this problem has three positive solutions, whereq∗ = 2∗/(2∗ − q) and f+ = max f , 0 6= 0.

To state our main result, we introduce precise conditions on V(x), f (x) and g(x):

(V) V(x) ∈ C(RN , R

), 0 < V0 := inf

x∈RNV(x) ≤ V(x) ≤ V∞ := lim

|x|→+∞V(x) < +∞,

( f ) f is positive, continuous and belongs to Lq∗ (RN), where q∗ is conjugate to p/q (i.e.q∗ = p/(p− q)),

(g) g(x) ∈ C(RN) ∩ L∞ (RN), 0 < g∞ := lim

|x|→+∞g(x) ≤ g(x) ≤ sup

x∈RNg(x) < +∞.

Our main result is as follows.Let σ := (p− 2)(2− q)(2−q)/(p−2)( Sp

p−q

)(p−q)/(p−2) and 0 < σ∗ = qσ/2 < σ, where Sp is thebest Sobolev constant described in the following Lemma 2.2.

Multiple positive solutions for Schrödinger problems 3

Theorem 1.1. Under the assumptions (V), ( f ) and (g), if | f |q∗ |g|(2−q)/(p−2)∞ ∈ (0, σ∗), Equation

(1.1) has at least two positive solutions, which correspond to negative energy and positive energy,respectively; in particular, the one with negative energy is a positive ground state solution.

The combined effects of a sub-linear and a super-linear terms change the structure of thesolution set. According to the behaviour of nonlinearities and to the results we want to prove,the method of the decomposition of Nehari manifold turns out to be more appropriate. Withthe help of suitable autonomous equation, the Ekeland variational principle and the theory ofLagrange multipliers, we can prove that Equation (1.1) has at least two positive solutions, oneof which is a positive ground state solution. In addition, the condition (V) can be replaced byother forms.

Remark 1.2. Assume that(V)

, ( f ) and (g), if | f |q∗ |g|(2−q)/(p−2)∞ is sufficiently small, then

Theorem 1.1 still holds.

Remark 1.3. Assume that V(x) ≡ C, ( f ) and (g), if | f |q∗ |g|(2−q)/(p−2)∞ is sufficiently small, then

Theorem 1.1 still holds, where C is a positive constant.

The rest of this paper is organized as follows: Section 2 is dedicated to our variationalframework and some preliminary results. Section 3 concerns with the proof of Theorem 1.1.

Throughout this paper, C and Ci denote distinct constants. Lp (RN) is the usual Lebesgue

space endowed with the standard norm |u|p =(∫

RN |u|pdx)1/p for 1 ≤ p < ∞ and |u|∞ =

supx∈RN |u(x)| for p = ∞. When it causes no confusion, we still denote by un a subsequenceof the original sequence un.

2 Preliminary results

With the fact that the problem (1.1) has a variational structure, the proof is based on thevariational approach and the use of the Nehari manifold technique. So, we will first recallsome preliminaries and establish the variational setting for our problem in this section.

Define

E :=

u ∈ H1(

RN)\0

∣∣∣∣ ∫RN

V(x)|u|2 dx < +∞

with the associate norm

‖u‖ =(∫

RN(|∇u|2 + V(x)u2)dx

) 12

.

Under the assumption (V), we know that the norm ‖ · ‖ is equivalent to the usual norm inH1 (RN). The energy functional corresponding to Equation (1.1) is

I(u) =12

∫RN

(|∇u|2 + V(x)|u|2

)dx− 1

q

∫RN

f (x)|u|q dx− 1p

∫RN

g(x)|u|p dx, u ∈ E. (2.1)

Lemma 2.1. If (V), ( f ) and (g) hold, then the functional I ∈ C1(E, R) and for any u, v ∈ E⟨I′(u), v

⟩=∫

RN∇u∇v dx +

∫RN

V(x)uv dx

−∫

RNf (x)|u|q−2uv dx−

∫RN

g(x)|u|p−2uv dx. (2.2)

Furthermore, I′ is weakly sequentially continuous in E.

4 X. Cao and J. Xu

Proof. The proof is a direct computation. Here we omit details and refer to [23].

Lemma 2.2 ([23]). Under the assumption (V), the embedding E → Lp (RN) is continuous forp ∈ [2, 2∗]. Let

Sp = infu∈E\0

‖u‖2(∫RN |u|p dx

)2/p > 0,

then|u|p ≤ S−

12

p ‖u‖, ∀u ∈ E.

It is well-known that seeking a weak solution of Equation (1.1) is equivalent to finding acritical point of the corresponding functional I. In the following, we are devoted to findingthe critical point of the corresponding functional I.

As usual, some energy functional such as I in (2.1) is not bounded from below on E but,as we will see, is bounded from below on an appropriate subset of E and a minimizer on thisset (if it exists) may give rise to a solution of corresponding differential equation (see [22]). Agood exemplification for an appropriate subset of E is the so-called Nehari manifold

N :=

u ∈ H1(RN)\0 |⟨

I′(u), u⟩= 0

,

where 〈 , 〉 denotes the usual duality between E and E∗. It is clear to see that u ∈ N if andonly if for u ∈ H1 (RN) \0,

‖u‖2 =∫

RNf (x)|u|q dx +

∫RN

g(x)|u|p dx. (2.3)

Obviously,N contains all nontrivial solutions of Equation (1.1). Below, we shall use the Neharimanifold methods to find critical points for the functional I.

The Nehari manifold N is closely linked to the behavior of functions of the form Ku : t→I(tu) for t > 0. Such maps are known as fibering maps, which were introduced by Drábekand Pohozaev in [9]. For u ∈ E, let

Ku(t) = I(tu) =12

t2‖u‖2 − 1q

tq∫

RNf (x)|u|q dx− 1

ptp∫

RNg(x)|u|p dx;

K′u(t) = t‖u‖2 − tq−1∫

RNf (x)|u|q dx− tp−1

∫RN

g(x)|u|p dx;

K′′u(t) = ‖u‖2 − (q− 1)tq−2∫

RNf (x)|u|q dx− (p− 1)tp−2

∫RN

g(x)|u|p dx.

Lemma 2.3. Let u ∈ E and t > 0. Then tu ∈ N if and only if K′u(t) = 0, that is, the critical points ofKu(t) correspond to the points on the Nehari manifold. In particular, u ∈ N if and only if K′u(1) = 0.

Proof. The result is an immediate consequence of the fact:

K′u(t) =⟨

I′(tu), u⟩=

1t⟨

I′(tu), tu⟩

.

Thus, it is natural to split N into three parts corresponding to local minima, points ofinflection and local maxima. Accordingly, we define

N+= u ∈ N | K′′u(1) > 0, N 0= u ∈ N | K′′u(1) = 0 and N−= u ∈ N | K′′u(1) < 0.

Multiple positive solutions for Schrödinger problems 5

It is easy to see that

K′′u(1) = ‖u‖2 − (q− 1)∫

RNf (x)|u|q dx− (p− 1)

∫RN

g(x)|u|p dx. (2.4)

DefineΨ(u) = K′u(1) =

⟨I′(u), u

⟩= ‖u‖2 −

∫RN

f (x)|u|q dx−∫

RNg(x)|u|p dx. (2.5)

Then for u ∈ N ,(ddt

Ψ(tu)) ∣∣∣∣

t=1=⟨Ψ′(u), u

⟩=⟨Ψ′(u), u

⟩−⟨

I′(u), u⟩= K′′u(1)

= ‖u‖2 − (q− 1)∫

RNf (x)|u|q dx− (p− 1)

∫RN

g(x)|u|p dx.

For each u ∈ N , Ψ(u) = K′u(1) = 0. Thus, for each u ∈ N , we have

K′′u(1) = K′′u(1)− (q− 1)Ψ(u) = (2− q)‖u‖2 − (p− q)∫

RNg(x)|u|p dx (2.6)

andK′′u(1) = K′′u(1)− (p− 1)Ψ(u) = (2− p)‖u‖2 + (p− q)

∫RN

f (x)|u|q dx. (2.7)

In order to ensure the Nehari manifold N to be a C1-manifold, we need the followingproposition.

Proposition 2.4. Let σ := (p− 2)(2− q)(2−q)/(p−2)( Spp−q

)(p−q)/(p−2), where Sp is the best Sobolev

constant described in Lemma 2.2. If | f |q∗ |g|(2−q)/(p−2)∞ ∈ (0, σ), then the set N 0 = ∅.

Proof. Suppose, on the contrary, there exists a u ∈ N such that K′′u(1) = 0. By Lemma 2.2,∫RN

g(x)|u|p dx ≤ |g|∞S−p2

p ‖u‖p. (2.8)

Noting that 2 < p < 2∗, from (2.6) we have

(2− q)‖u‖2 ≤ (p− q)|g|∞S−p2

p ‖u‖p,

so

‖u‖ ≥

(2− q)Sp2p

(p− q)|g|∞

1p−2

. (2.9)

Moreover, by the Hölder inequality and Lemma 2.2, we have

∫RN

f (x)|u|q dx ≤(∫

RN| f (x)|q∗ dx

) 1q∗(∫

RN|u|p dx

) qp

= | f |q∗|u|qp ≤ | f |q∗S− q

2p ‖u‖q. (2.10)

From (2.7) we have

(p− 2)‖u‖2 ≤ (p− q)| f |q∗S− q

2p ‖u‖q,

which implies that

‖u‖ ≤

(p− q)| f |q∗

(p− 2)Sq2p

12−q

. (2.11)

6 X. Cao and J. Xu

This with (2.9) and (2.11) implies that

| f |q∗ |g|2−qp−2∞ >

(2− q)Sp2p

p− q

2−qp−2

p− 2p− q

Sq2p = (p− 2)(2− q)

2−qp−2

(Sp

p− q

) p−qp−2

= σ,

which contradicts with the condition.

Proposition 2.5. Suppose that | f |q∗ |g|(2−q)/(p−2)∞ ∈ (0, σ) and u ∈ E. Then, there are unique t+ and

t− with 0 < t+ < tmax < t− such that t+u ∈ N+, t−u ∈ N− and

I(t+u) = inf0≤t≤tmax

I(tu), I(t−u) = supt≥tmax

I(tu).

Proof. Let

h(t) = t2−q‖u‖2 − tp−q∫

RNg(x)|u|p dx,

then we have

K′u(t) = tq−1(

h(t)−∫

RNf (x)|u|q dx

). (2.12)

Clearly, h(0) = 0 and h(t)→ −∞ as t→ ∞. From 1 < q < 2 < p < 2∗ and

h′(t) = tp−q−1((2− q)t2−p‖u‖2 − (p− q)

∫RN

g(x)|u|p dx)= 0,

we can infer that there is a unique tmax > 0 such that h(t) achieves its maximum at tmax,increasing for t ∈ [0, tmax) and decreasing for t ∈ (tmax, ∞) with limt→∞ h(t) = −∞, where

tmax =

((2− q)‖u‖2

(p− q)∫

RN g(x)|u|p dx

) 1p−2

.

It follows

h(tmax) = ‖u‖q

(‖u‖p∫

RN g(x)|u|pdx

) 2−qp−2 ( 2− q

p− q

) 2−qp−2 p− 2

p− q

≥ ‖u‖q

‖u‖p

|g|∞S−p2

p ‖u‖p

2−qp−2 (

2− qp− q

) 2−qp−2 p− 2

p− q

= ‖u‖q

(2− q)Sp2p

|g|∞(p− q)

2−qp−2

p− 2p− q

> 0.

(2.13)

From | f |q∗ |g|(2−q)/(p−2)∞ ∈ (0, σ), (2.10) and (2.13) we also have

∫RN

f (x)|u|q dx < ‖u‖q

(2− q)Sp2p

|g|∞(p− q)

2−qp−2

p− 2p− q

< h (tmax) . (2.14)

Moreover, for tu ∈ N , K′u(t) = 0. By (2.12) we obtain that

K′′u(t) = tq−1h′(t).

Multiple positive solutions for Schrödinger problems 7

By (2.12) and (2.14) we know there are unique t+ and t− with 0 < t+ < tmax < t− such thatK′t+u(1) = 0, K′t−u(1) = 0, that is t+u, t−u ∈ N . From K′′u(t) = tq−1h′(t) and h′(t+) > 0 >

h′(t−), one arrives at the conclusion.

The forthcoming lemma is to obtain the minimizing sequence of the energy functional Ion the Nehari manifold N .

Lemma 2.6. The energy functional I is coercive and bounded from below on N .

Proof. For u ∈ N , then, by the Hölder inequality and Lemma 2.2,

I(u) = I(u)− 1p⟨

I′(u), u⟩

=

(12− 1

p

)‖u‖2 −

(1q− 1

p

) ∫RN

f (x)|u|q dx

≥(

12− 1

p

)‖u‖2 −

(1q− 1

p

)| f |q∗S

− q2

p ‖u‖q.

This completes the proof.

Lemma 2.7. Under the assumptions (V), ( f ) and (g), the following results hold.

(i) If | f |q∗ |g|(2−q)/(p−2)∞ ∈ (0, σ), then c1 = infu∈N+ I(u) < 0;

(ii) If | f |q∗ |g|(2−q)/(p−2)∞ ∈ (0, σ∗), then c2 = infu∈N− I(u) > 0, where σ∗ = qσ/2 and σ described

in Proposition 2.4.

Proof. (i) For each u ∈ N+, K′′u(1) > 0. From (2.7), we have

(p− q)∫

RNf (x)|u|q dx > (p− 2)‖u‖2.

If | f |q∗ |g|(2−q)/(p−2)∞ ∈ (0, σ), then

I(u) = I(u)− 1p⟨

I′(u), u⟩=

p− 22p‖u‖2 − p− q

pq

∫RN

f (x)|u|q dx

<p− 2

2p‖u‖2 − p− 2

pq‖u‖2 =

(p− 2)(q− 2)2pq

‖u‖2 < 0.(2.15)

Thus, infu∈N+ I(u) < 0.(ii) For each u ∈ N−, K′′u(1) < 0. From (2.9) and (2.10), we have if | f |q∗ |g|(2−q)/(p−2)

∞ ∈(0, σ∗), then

I(u) = I(u)− 1p⟨

I′(u), u⟩=

(12− 1

p

)‖u‖2 −

(1q− 1

p

) ∫RN

f (x)|u|q dx

≥(

12− 1

p

)‖u‖2 −

(1q− 1

p

)| f |q∗S

− q2

p ‖u‖q

= ‖u‖q((

12− 1

p

)‖u‖2−q −

(1q− 1

p

)| f |q∗S

− q2

p

)

≥

(2− q)Sp2p

(p− q)|g|∞

q

p−2(1

2− 1

p

) (2− q)Sp2p

(p− q)|g|∞

2−qp−2

−(

1q− 1

p

)| f |q∗S

− q2

p

> 0.

8 X. Cao and J. Xu

Lemma 2.8. If | f |q∗ |g|(2−q)/(p−2)∞ ∈ (0, σ), then the set N− is closed in E.

Proof. Let un ⊂ N− such that un → u in E. In the following we prove u ∈ N−. Indeed, forany u ∈ N−, from (2.6) we have

(2− q)‖u‖2 < (p− q)∫

RNg(x)|u|p dx.

Similar to the proof of (2.9), we have

‖u‖ ≥

(2− q)Sp2p

(p− q)|g|∞

1p−2

. (2.16)

Hence N− is bounded away from 0.By 〈I′ (un) , un〉 = 0 and Lemma 2.1, we have 〈I′(u), u〉 = 0. (2.6) implies that K′′un

(1) →K′′u(1). From K′′un

(1)< 0, we have K′′u(1) ≤ 0. By Proposition 2.4 we know, if | f |q∗ |g|(2−q)/(p−2)∞ ∈

(0, σ), then K′′u(1) < 0. Thus we deduce u ∈ N−.

The following lemma is used to extract a (PS)c1 (or (PS)c2) sequence from the minimizingsequence of the energy functional I on the Nehari manifold N+ (or N−).

Lemma 2.9. If | f |q∗ |g|(2−q)/(p−2)∞ ∈ (0, σ), then for every u ∈ N+, there exist ε > 0 and a differen-

tiable function ϕ+ : Bε(0)→ R+ := (0,+∞) such that

ϕ+(0) = 1, ϕ+(w)(u− w) ∈ N+, ∀w ∈ Bε(0)

and〈(ϕ+)′(0), w〉 = L(u, w)/K′′u(1), (2.17)

whereL(u, w) = 2〈u, w〉 − q

∫RN

f (x)|u|q−2uw dx− p∫

RNg(x)|u|p−2uw dx.

Moreover, for any C1, C2 > 0, there exists C > 0 such that if C1 ≤ ‖u‖ ≤ C2, then∣∣∣⟨(ϕ+)′(0), w

⟩∣∣∣ ≤ C‖w‖.

Proof. We define F : R× E→ R by

F(t, w) = K′u−w(t),

it is easy to see that F is differentiable. Since F(1, 0) = 0 and

Ft(1, 0) = K′′u(1) > 0,

we apply the implicit function theorem at point (1, 0) to obtain the existence of ε > 0 anddifferentiable function ϕ+ : Bε(0)→ R+ := (0,+∞) such that

ϕ+(0) = 1, F(

ϕ+(w), w)= 0, ∀w ∈ Bε(0).

Thus,ϕ+(w)(u− w) ∈ N , ∀w ∈ Bε(0).

Multiple positive solutions for Schrödinger problems 9

Next, we prove for any w ∈ Bε(0), ϕ+(u − w) ∈ N+. Indeed, by u ∈ N+ and the setN− ∪N 0 is closed, we know dist

(u,N− ∪N 0) > 0. Since ϕ+(w)(u− w) is continuous with

respect to w, we know when ε is small enough, for w ∈ Bε(0), then∥∥ϕ+(w)(u− w)− u∥∥ <

12

dist(u,N− ∪N 0) ,

so ∥∥ϕ+(w)(u− w)−N− ∪N 0∥∥ ≥ dist(u,N− ∪N 0)− dist

(ϕ+(w)(u− w), u

)>

12

dist(u,N− ∪N 0) > 0.

Thus, for w ∈ Bε(0), then ϕ+(w)(u− w) ∈ N+.Besides, by the differentiability of implicit function theorem, we have

⟨(ϕ+)′(0), w

⟩= −〈Fw(1, 0), w〉

Ft(1, 0).

Note that L(u, w) = − 〈Fw(1, 0), w〉 and K′′u(1) = Ft(1, 0). Therefore (2.17) holds.In the following we prove that there exists δ > 0 such that K′′u(1) ≥ δ > 0 with C1 ≤ ‖u‖ ≤

C2, u ∈ N+, where C1, C2 > 0. On the contrary, if there exists a sequence un ∈ N+ withC1 ≤ ‖un‖ ≤ C2, such that for any δn sufficiently small, K′′un

(1) ≤ δn, δn → 0 as n → ∞. From(2.6) we have

(2− q) ‖un‖2 = (p− q)∫

RNg(x) |un|p dx + O(δn),

where O(δn) → 0 as δn → 0. Noting that 1 < q < 2 < p < 2∗, C1 ≤ ‖un‖ ≤ C2 and (2.8), wehave

(2− q) ‖un‖2 ≤ (p− q)|g|∞S−p2

p ‖un‖p + O (δn) ,

and so

‖un‖ ≥((2− q)Sp/2

p

(p− q)|g|∞

)1/(p−2)

+ O (δn) . (2.18)

From (2.7) we also have

(p− 2) ‖un‖2 = (p− q)∫

RNf (x) |un|q dx + O (δn) .

In view of (2.10), we have

(p− 2) ‖un‖2 ≤ (p− q)| f |q∗S− q

2p ‖un‖q + O (δn) ,

which implies that

‖un‖ ≤((p− q)| f |q∗(p− 2)Sq/2

p

)1/(2−q)

+ O (δn) . (2.19)

Let n→ ∞, from (2.18) and (2.19) we deduce a contradiction.Thus if C1 ≤ ‖u‖ ≤ C2, then there exists C > 0 such that∣∣∣⟨(ϕ+

)′(0), w

⟩| ≤ C‖w‖.

This completes the proof.

10 X. Cao and J. Xu

Similarly, we establish the following lemma.

Lemma 2.10. If | f |q∗ |g|(2−q)/(p−2)∞ ∈ (0, σ), then for every u ∈ N−, there exist ε > 0 and a

differentiable function ϕ− : Bε(0)→ R+ := (0,+∞)

ϕ−(0) = 1, ϕ−(w)(u− w) ∈ N−, ∀w ∈ Bε(0)

and ⟨(ϕ−)′(0), w

⟩= L(u, w)/K′′u(1),

whereL(u, w) = 2〈u, w〉 − q

∫RN

f (x)|u|q−2uw dx− p∫

RNg(x)|u|p−2uw dx.

Moreover, for any C1, C2 > 0, there exists C > 0 such that if C1 ≤ ‖u‖ ≤ C2,

|〈(ϕ−)′(0), w〉| ≤ C‖w‖.

From above, we can extract a (PS)c1 (or (PS)c2) sequence from the minimizing sequenceof the energy functional I on the Nehari manifold N+ (or N−).

Lemma 2.11. If | f |q∗ |g|(2−q)/(p−2)∞ ∈ (0, σ), then the minimizing sequence un ⊂ N+ is the (PS)c1

sequence in E.

Proof. By Lemma 2.10 and the Ekeland Variational Principle [10, 23] on N+ ∪N 0, there existsa minimizing sequence un ⊂ N+ ∪N 0 such that

infu∈N+∪N 0

I(u) ≤ I (un) < infu∈N+∪N 0

I(u) +1n

, (2.20)

I (un)−1n‖v− un‖ ≤ I(v), ∀v ∈ N+ ∪N 0. (2.21)

From Proposition 2.5, we know for each u ∈ E\0, there is a unique t+ such that t+u ∈N+, then infu∈N+ I ≤ I (t+u). By Lemma 2.7 and I(0) = 0, we get that infu∈N+∪N 0 I(u) =

infu∈N+ I(u) = c1. Thus we may assume un ∈ N+, I (un) → c1 < 0. By Lemma 2.9, since| f |q∗ |g|(2−q)/(p−2)

∞ ∈ (0, σ), we can find εn > 0 and differentiable function ϕ+n = ϕ+

n (w) > 0such that

ϕ+n (w) (un − w) ∈ N+, ∀w ∈ Bεn(0).

By the continuity of ϕ+n (w) and ϕ+

n (0) = 1, without loss of generality, we can assume εn issufficiently small such that 1/2 ≤ ϕ+

n (w) ≤ 3/2 for ‖w‖ < εn. From ϕ+n (w) (un − w) ∈ N+

and (2.21) , we have

I(

ϕ+n (w) (un − w)

)≥ I (un)−

1n∥∥ϕ+

n (w) (un − w)− un∥∥ ,

which implies that⟨I′ (un) , ϕ+

n (w) (un − w)− un⟩+ o

(∥∥ϕ+n (w) (un − w)− un

∥∥) ≥ − 1n∥∥ϕ+

n (w) (un − w)− un∥∥ .

Consequently,

ϕ+n (w)

⟨I′ (un) , w

⟩+(1− ϕ+

n (w)) ⟨

I′ (un) , un⟩

≤ 1n∥∥(ϕ+

n (w)− 1)

un − ϕ+n (w)w

∥∥+ o(∥∥ϕ+

n (w) (un − w)− un∥∥) .

Multiple positive solutions for Schrödinger problems 11

By the choice of εn and 1/2 ≤ ϕ+n (w) ≤ 3/2, we infer that there exists C3 > 0 such that

∣∣⟨I′ (un) , w⟩∣∣ ≤ 1

n

∥∥∥⟨(ϕ+n)′(0), w

⟩un

∥∥∥+ C3

n‖w‖+ o

(∣∣∣⟨(ϕ+n)′(0), w

⟩∣∣∣ (‖un‖+ ‖w‖))

.

Below we prove for un ⊂ N+, infn ‖un‖ ≥ C1 > 0, where C1 is a constant. Indeed, ifnot, then I (un) would converge to zero, which contradicts I (un) → c1 < 0. Moreover, byLemma 2.6 we know that I is coercive on N+, un is bounded in E. Thus, there exists C2 > 0such that 0 < C1 ≤ ‖un‖ ≤ C2. From Lemma 2.9,

∣∣⟨(ϕ+n )′(0), w

⟩∣∣ ≤ C‖w‖. So

∣∣⟨I′ (un) , w⟩∣∣ ≤ C

n‖w‖+ C

n‖w‖+ o(‖w‖)

and ∥∥I′ (un)∥∥ = sup

w∈E\0

|〈I′ (un) , w〉|‖w‖ ≤ C

n+ o(1),∥∥I′ (un)

∥∥→ 0, as n→ ∞.

(2.22)

Thus, un ⊂ N+ is (PS)c1 for I in E.

Lemma 2.12. If | f |q∗ |g|(2−q)/(p−2)∞ ∈ (0, σ), then the minimizing sequence un ⊂ N− is the (PS)c2

sequence in E.

Proof. From Lemma 2.8, N− is closed in E. By Lemma 2.6, we know I is coercive on N−.So we use the Ekeland Variational Principle [23] on N− to obtain a minimizing sequenceun ⊂ N− such that

infu∈N−

I(u) ≤ I (un) < infu∈N−

I(u) +1n

,

I (un)−1n‖v− un‖ ≤ I(v), ∀v ∈ N−.

In view of (2.15) and Lemma 2.6, we know that there exist C1, C2 > 0 such that

0 < C1 ≤ ‖un‖ ≤ C2.

Hence by Lemma 2.10, in the same way as Lemma 2.11, there exists a minimizing sequenceun ⊂ N− is the (PS)c2 sequence in E.

The following lemmas aims at obtaining the critical points of I on the whole space fromthe critical points of I|N+ and I|N− , respectively.

Lemma 2.13. Suppose that u is a local minimizer for I on N+. Then I ′(u) = 0.

Proof. If u 6= 0, u is a local minimizer for I on N+, then u is a nontrivial solution of theoptimization problem

minimize I subject to Ψ(u) = 0,

where Ψ(u) is described in (2.5). Then, u ∈ N+ ⊂ N such that

I(u) = c1 = infu∈N+

I(u) = infu∈N

I(u).

12 X. Cao and J. Xu

Note that Ψ′(u) 6= 0 and N+ is a local differential manifold. So by the theory of Lagrangemultipliers, there exists µ ∈ R such that I′(u) = µΨ′(u). Thus⟨

I′(u), u⟩= µ

⟨Ψ′(u), u

⟩.

Since u ∈ N+, we have 〈I′(u), u〉 = 0 and 〈Ψ′(u), u〉 = K′′u(1) 6= 0. Hence, µ = 0 andI′(u) = 0.

Lemma 2.14. Suppose that u is a nontrivial critical point of I|N− , then it is a nontrivial critical pointof I in E, i.e., I′(u) = 0.

Proof. If u is a nontrivial critical point of I|N− , i.e., u ∈ N−\0 and (I|N−)′(u) = 0. Notethat N− is a local differential manifold and Ψ′(u) 6= 0, where Ψ(u) is described in (2.5). Soby the theory of Lagrange multipliers, there exists µ ∈ R such that I′(u) = µΨ′(u). Thus⟨

I′(u), u⟩= µ

⟨Ψ′(u), u

⟩.

Since u ∈ N−, we have 〈I′(u), u〉 = 0 and 〈Ψ′(u), u〉 = K′′u(1) 6= 0. Hence, µ = 0 and I′(u) = 0.Thus the proof is complete.

3 Proof of Theorem 1.1

In order to obtain the nontrivial solutions, we bring in the following lemma.

Lemma 3.1 (Lions [19, 20, 23]). Let r > 0, q ∈ [2, 2∗). If un is bounded in H1(RN) and

limn→∞

supy∈RN

∫Br(y)|u|q dx = 0,

then we have un → 0 in Lp (RN) for p ∈ (2, 2∗). Here 2∗ = 2N/(N − 2) if N ≥ 3 and 2∗ = ∞ ifN = 1, 2.

Lemma 3.2. Let un ⊂ E be a bounded (PS)c sequence for I. Then either

(i) un → 0 in E, or

(ii) there exist a sequence yn ∈ RN and constants r, δ > 0 such that

lim infn→∞

∫Br(yn)

|un|2 dx ≥ δ > 0.

Proof. Suppose the condition (ii) is not satisfied, i.e. for any r > 0, we have

limn→∞

supy∈RN

∫Br(y)|un|2 dx = 0.

Then by Lemma 3.1, un → 0 in Lp(RN) for p ∈ (2, 2∗) . Therefore,

0 ≤∣∣∣∣∫

RNf (x) |un|q dx +

∫RN

g(x) |un|p dx∣∣∣∣ ≤ | f |q∗ |un|qp + |g|∞ |un|pp → 0.

Since un ⊂ E is a bounded (PS)c sequence for I, we have

o(1) = I′ (un) un =∫

RN

(|∇un|2 + V(x) |un|2

)dx−

(∫RN

f (x) |un|q dx +∫

RNg(x) |un|p dx

),

as n → ∞. It follows that un → 0 in E as n → ∞, i.e., the condition (i) is satisfied. Thus, theproof is complete.

Multiple positive solutions for Schrödinger problems 13

To recover the compactness, we need to evaluate the critical value of Equation (1.1) throughthe critical value of a autonomous equation. Now, we consider the following autonomousequation

−4u + V∞u = g∞|u|p−2u in RN ,

u ∈ H1(RN),(3.1)

where 2 < p < 2∗(2∗ = ∞ if N = 1, 2 and 2∗ = 2N/(N − 2) if N ≥ 3). The correspondingfunctional and the corresponding manifold are

I∞(u) =12

∫RN

(|∇u|2 + V∞|u|2

)dx− 1

p

∫RN

g∞|u|p dx

andN∞ =

u ∈ H1

(RN)\ 0

∣∣∣ ⟨I′∞(u), u⟩= 0

.

Let w0 be the unique radially symmetric solution of Equation (3.1) such that I∞ (w0) = c∞,where c∞ = infu∈N∞ I∞(u) (see [3, 17]).

In the following, we prove that when the critical value of Equation (1.1) is contained in thesuitable range, (PS)c condition holds.

Proposition 3.3. Let the assumptions of (V), ( f ) and (g) be satisfied, if | f |q∗ |g|(2−q)/(p−2)∞ ∈ (0, σ∗),

then each (PS)c sequence un ⊂ N (N = N+ or N−) for I in E with c < c1 + c∞ has a stronglyconvergent subsequence, where c1 is described in Lemma 2.7.

Proof. Let un ⊂ N such that

I (un)→ c and I′ (un)→ 0 as n→ ∞.

From Lemma 2.6 we know that the (PS)c sequence un ⊂ N for I in E is bounded. Then,going if necessary to a subsequence, we have

un u in E,

un → u in Lrloc(R

N), r ∈ [2, 2∗) ,

un → u a.e. in RN .

(3.2)

Set vn := un − u, then there exists C > 0 such that ‖vn‖ < C. It is sufficient to prove thatvn → 0 in E as n→ ∞.

Note that ∣∣|un|s − |u|s∣∣ ≤ |un − u|s for s > 1, (3.3)

we can infer that∫RN

f (x) |un|q dx →∫

RNf (x)|u|q dx and

∫RN

f (x) |vn|q dx → 0 as n→ ∞. (3.4)

Indeed, from the condition ( f ), we have that for any ε > 0, there exists R sufficiently largesuch that (∫

|x|>R| f (x)|q∗ dx

)1/q∗

< ε.

14 X. Cao and J. Xu

And from un ⊂ N in E is bounded, we can infer that(∫|x|>R |un − u|p dx

)q/pis bounded.

These facts with (3.2), we have∫RN

∣∣ f (x)(|un|q − |u|q

)∣∣ dx ≤∫

RNf (x) |un − u|q dx

=∫|x|≤R

f (x) |un − u|q dx +∫|x|>R

f (x) |un − u|q dx

≤(∫|x|≤R

| f (x)|q∗ dx)1/q∗ (∫

|x|≤R|un − u|p dx

)q/p

+

(∫|x|>R

| f (x)|q∗ dx)1/q∗ (∫

|x|>R|un − u|p dx

)q/p

→ 0 as n→ ∞.

From (3.2) and Brézis–Lieb lemma in [23], we can deduce that

I (vn) = I (un − u)

=12

∫RN

(|∇ (un − u)|2 + V(x) |un − u|2

)dx

− 1p

∫RN

g(x) |un − u|p dx− 1q

∫RN

f (x) |un − u|q dx

= I (un)− I(u) + o(1)

(3.5)

and

I′ (vn) vn = I′ (un − u) (un − u)

=∫

RN

(|∇ (un − u)|2 + V(x) |un − u|2

)dx

−∫

RNf (x) |un − u|q dx−

∫RN

g(x) |un − u|p dx.

= I′ (un) un − I′(u)u + o(1).

(3.6)

By Lemma 2.1, I′ is weakly sequentially continuous in E, so I′(u) = 0. Therefore if u 6= 0and I′(u)u = 0, then u ∈ N+ or u ∈ N−. According to Lemma 2.7, no matter u ∈ N+ oru ∈ N−, we all have I(u) ≥ c1. If u = 0, then I(u) = I(0) = 0 > c1. So

I (vn) = I (un)− I(u) + o(1) ≤ c− c1 + o(1) (3.7)

andI′ (vn) vn = o(1). (3.8)

Indeed, if vn 9 0 in E, we choose (tn) ⊂ (0, ∞) such that tnvn ⊂ N∞. We will provethat the case of lim supn→∞ tn > 1, lim supn→∞ tn < 1 and lim supn→∞ tn = 1 cannot happen.Then we obtain a contradiction and vn → 0 in E. To do this, we distinguish the following threecases:

(i) lim supn→∞ tn > 1.

In this case, we may suppose there exist σ > 0 and a subsequence still denoted by tn suchthat tn ≥ 1 + σ for all n ∈N. From (3.6) and (3.8), we have

I′ (vn) vn =∫

RN

(|∇vn|2 + V(x) |vn|2

)dx−

∫RN

f (x)|vn|q dx−∫

RNg(x) |vn|p dx = o(1). (3.9)

Multiple positive solutions for Schrödinger problems 15

Moreover, since tnvn ⊂ N∞, then we have

I′∞ (tnvn) tnvn = t2n

∫RN

(|∇vn|2 + V∞ |vn|2

)dx− tp

n

∫RN

g∞ |vn|p dx = 0. (3.10)

Combining (3.9) and (3.10), we obtain that∫RN

(V∞ −V(x)) |vn|2 dx +∫

RN(g(x)− g∞) |vn|p dx +

∫RN

f (x) |vn|q dx

=∫

RNg∞

(tp−2n − 1

)|vn|p dx + o(1).

By conditions (V) and (g), for any ε > 0, there exists R = R(ε) > 0 such that

V(x) ≥ V∞ − ε and g∞ ≥ g(x)− ε for any |x| > R. (3.11)

This with (3.2) and (3.4) implies that((1 + σ)p−2 − 1

) ∫RN

g∞ |vn|p dx ≤ Cε + o(1). (3.12)

By vn 9 0 in E and (3.9), similar to Lemma 3.2, we can prove that there exist a sequenceyn ⊂ RN and constants r, δ > 0 such that

lim infn→∞

∫Br(yn)

|vn|2 dx ≥ δ > 0. (3.13)

If we set wn(x) = vn (x + yn) , then there exists a function w and a subsequence stilldenoted by wn such that wn w in E, wn → w in Ls

loc

(RN) where s ∈ [2, 6) and wn(x) →

w(x) a.e. in RN . Moreover, by (3.13) there exists a subset Λ ⊂ RN with positive measure suchthat w 6= 0 a.e. in Λ. It follows from (3.12) that

0 <((1 + σ)p−2 − 1

) ∫Λ

g∞|w|p dx ≤ Cε + o(1),

where ε > 0 is arbitrary. This is impossible.

(ii) lim supn→∞ tn < 1.

In this case, without loss of generality, we suppose that tn < 1 for all n ∈ N. From (3.2),(3.4), (3.7), (3.9), (3.10) and (3.11), we can deduce that

c∞ ≤ I∞ (tnvn) = I∞ (tnvn)−1p⟨

I′∞ (tnvn) , tnvn⟩

=

(12− 1

p

) ∫RN

(|∇tnvn|2 + V∞ |tnvn|2

)dx <

(12− 1

p

) ∫RN

(|∇vn|2 + V∞ |vn|2

)dx

=

(12− 1

p

) ∫RN

(|∇vn|2 + V(x) |vn|2 + (V∞ −V(x)) |vn|2

)dx

=12

∫RN

(|∇vn|2 + V(x) |vn|2

)dx− 1

p

∫RN

g(x) |vn|p dx

+

(12− 1

p

)(V∞ −V(x)) |vn|2 dx + o(1)

≤ 12

∫RN

(|Oun|2 + V(x) |un|2

)dx− 1

p

∫RN

g(x) |un|p dx− 1q

∫RN

f (x) |un|q dx

−(

12

∫RN

(|Ou|2 + V(x)|u|2

)dx− 1

p

∫RN

g(x)|u|p dx− 1q

∫RN

f (x)|u|q dx)+ Cε + o(1)

= I (un)− I(u) + Cε + o(1) ≤ c− c1 + Cε + o(1).

Let n→ ∞, we get c ≥ c1 + c∞. This contradicts c < c1 + c∞.

16 X. Cao and J. Xu

(iii) lim supn→∞ tn = 1.

In this case, there exists a subsequence, still denoted by tn such that tn → 1 as n → ∞.Note that

I (vn)− I∞ (tnvn) =12

∫RN

(1− t2

n)|∇vn|2 dx +

12

∫RN

V(x) |vn|2 dx− t2n2

∫RN

V∞ |vn|2 dx

− 1p

∫RN

g(x) |vn|p dx +1p

∫RN

g∞ |tnvn|p dx− 1q

∫RN

f (x) |vn|q dx.

From (3.2), (3.4) and (3.11), we can infer that

I (vn) ≥ I∞ (tnvn)− Cε + o(1) ≥ c∞ − Cε + o(1).

This with (3.7) implies that c ≥ c1 + c∞, which contradicts c < c1 + c∞.

Lemma 3.4. If | f |q∗ |g|(2−q)/(p−2)∞ ∈ (0, σ∗), Equation (1.1) has at least one positive solution.

Proof. From Lemma 2.11, we know if | f |q∗ |g|(2−q)/(p−2)∞ ∈ (0, σ∗), then there is minimizing

sequence un ⊂ N+ which is a (PS)c1 sequence in E. Obviously, c1 < c1 + c∞, so fromProposition 3.3, there is a strongly convergent subsequence still denoted by un such thatun → u1 in E. From Lemma 2.11 we know there exist C1, C2 > 0 such that 0 < C1 ≤ ‖un‖ ≤ C2,then 0 < C1 ≤ ‖u1‖ ≤ C2. Thus u1 6= 0.

Next we prove u1 ∈ N+. Indeed, By (2.6), it follows that K′′un(1)→ K′′u1

(1). From K′′un(1) >

0, we have K′′u1(1) ≥ 0. By Proposition 2.4 and u1 6= 0 we know, if | f |q∗ |g|(2−q)/(p−2)

∞ ∈ (0, σ∗),then K′′u1

(1) > 0. Thus

u1 ∈ N+, I (u1) = limn→∞

I (un) = infu∈N+

I(u).

We recall (see [11]) that∫

RN |∇|u||2dx =∫

RN |∇u|2dx, therefore I (u1) = I (|u1|) and |u1| ∈N+, then, without loss of generality, we may assume that u1 is positive. This with Lemma 2.13implies the desired result.

In the following, motivated by the arguments in [26], we will prove c2 < c1 + c∞. Let

wl(x) = w0(x + le), for l ∈ R and e ∈ SN−1,

where SN−1 =

x ∈ RN : |x| = 1

. Then, wl(x) is also a positive solution of limit equation(3.1) and I′∞ (wl)wl = 0, c∞ = I∞ (wl).

Lemma 3.5. Under the assumptions of Proposition 3.3, if | f |q∗ |g|(2−q)/(p−2)∞ ∈ (0, σ∗), then

c2 < c1 + c∞.

Proof. We prove this result in the following two steps.Step 1: For all l ∈ R, supt≥0 I (u1 + twl) < c1 + c∞.Since

I (u1 + twl)→ I (u1) = c1 < 0 as t→ 0

andI (u1 + twl)→ −∞ as t→ ∞,

Multiple positive solutions for Schrödinger problems 17

then there exist t2 > t1 > 0 such that I (u1 + twl) < c1 + c∞ for all t ∈ [0, t1] ∪ [t2, ∞). It issufficient to prove that supt1≤t≤t2

I (u1 + twl) < c1 + c∞. Indeed, by Willem [23], we know that

I∞ (twl) ≤ c∞ for all l ∈ R.

Note that

(u + v)p − up − vp − pup−1v ≥ 0 for (u, v) ∈ [0, ∞)× [0, ∞) and p > 2.

Furthermore, since u1 is one of positive solution of Equation (1.1), wl(x) is a positive solutionof limit equation (3.1), t1 ≤ t ≤ t2 and the conditions (V), ( f ), (g), we can infer that

I (u1 + twl) =12

∫RN|∇ (u1 + twl)|2 dx +

12

∫RN

V(x) |u1 + twl |2 dx

− 1q

∫RN

f (x) |u1 + twl |q dx− 1p

∫RN

g(x) |u1 + twl |p dx

≤ 12

∫RN|∇u1|2 dx +

12

∫RN|∇twl |2 dx +

∫RN

t∇u1∇wl dx +12

∫RN

V(x) |u1|2 dx

+12

∫RN

V∞ |twl |2 dx +12

∫RN

(V(x)−V∞) |twl |2 dx

+∫

RNtV(x)u1wl dx− 1

q

∫RN

f (x) |u1|q dx

− 1p

(∫RN

g(x) |u1|p dx +∫

RNg∞ |twl |p dx ++p

∫RN

g(x) |u1|p−1 twl dx)

− 1p

∫RN

(g(x)− g∞) |twl |p dx

< I (u1) + I∞ (twl) ≤ c1 + c∞.

Step 2: There exist t0 > 0, sl ∈ (0, 1) such that u1 + slt0wl ∈ N−, then combining Step 1, weobtain c2 < c1 + c∞.

First, we prove that

N− =

u ∈ E\0 :

1‖u‖ t−

(u‖u‖

)= 1

.

Indeed, for u ∈ N−, set v = u/‖u‖, then by Proposition 2.5, there is a unique t−(v) > 0 suchthat t−(v)v ∈N− i.e. t− (u/‖u‖) u/‖u‖ ∈N−. Because of the uniqueness, t− (u/‖u‖) 1/‖u‖=1 is proved. For u ∈ E\0 with t− (u/‖u‖) 1/‖u‖ = 1, set v = u/‖u‖ ∈ E\0, then byProposition 2.5, there is a unique t−(v) > 0 such that t−(v)v = t− (u/‖u‖) u/‖u‖ ∈ N−, sou ∈ N−. Let

U1 =

u ∈ E :

1‖u‖ t−

(u‖u‖

)> 1

∪ 0

and

U2 =

u ∈ E :

1‖u‖ t−

(u‖u‖

)< 1

.

Then N− separates E into two connected components U1 and U2, that is E \ N− = U1 ∪U2.Define a path γl(s) = u1 + st0wl for s ∈ [0, 1], then γl(0) = u1 and γl(1) = u1 + t0wl . If

we can prove γl(0) = u1 ∈ U1 and γl(1) = u1 + t0wl ∈ U2, the continuity of t(u) as in [23]yields that there exists sl ∈ (0, 1) such that u1 + slt0wl ∈ N−. Thus, it is sufficient to provethat (i) γl(0) = u1 ∈ U1 and (ii) γl(1) = u1 + t0wl ∈ U2.

18 X. Cao and J. Xu

(i) γl(0) = u1 ∈ U1.

Indeed, u1 ∈ N+, if N+ ⊂ U1, then u1 ∈ U1. In the following we prove N+ ⊂ U1. Forany u ∈ N+ ⊂ E, there is unique t+(u) such that t+(u)u ∈ N+. From the uniqueness,we obtain t+(u) = 1. By Proposition 2.5, we have 1 = t+(u) < tmax(u) < t−(u). Sincet−(u) = t− (u/‖u‖) /‖u‖, then 1 < t− (u/‖u‖) /‖u‖, that is N+ ⊂ U1.

(ii) γl(1) = u1 + t0wl ∈ U2.

Indeed, for any un ∈ E \ 0, there exists t−n := t− (un) such that t−n un ⊂ N−, we fist provet−n is bounded. Suppose on the contrary that there exists a subsequence, we still denotet−n , such that t−n → ∞. Then I (t−n un)→ −∞, this contradicts Lemma 2.6 I is bounded frombelow on N−. So there exists M > 0 such that t− ((u1 + t0wl) / ‖u1 + t0wl‖) < M. Let

t0 =

(p− 2pc∞

∣∣∣M2 − ‖u1‖2∣∣∣) 1

2

,

where

c∞ = I∞ (wl) = I∞ (wl)−1p

I′∞ (wl)wl =

(12− 1

p

) ∫RN

(|∇wl |2 + V∞ |wl |2

)dx.

Since wl(x) = w0(x + le) 0, ∇wl = ∇w0(x + le) 0 as l → ∞ and V is a positive boundedfunction, we have

∫|x|≤R (V∞ −V(x)) |wl |2 dx → 0 and

∫|x|>R (V∞ −V(x)) |wl |2 dx → 0 as

l → ∞ due to (3.11). Then

‖u1 + t0wl‖2 = ‖u1‖2 + t20 ‖wl‖2 + 2t0

∫RN

(∇u1∇wl + V(x)u1wl) dx

= ‖u1‖2 +p− 2pc∞

‖wl‖2∣∣∣M2 − ‖u1‖2

∣∣∣+ 2t0

∫RN

(∇u1∇wl + V(x)u1wl) dx

= ‖u1‖2 +p− 2pc∞

‖wl‖2∣∣∣M2 − ‖u1‖2

∣∣∣+ o(1) as l → ∞.

and

c∞ = I∞ (wl) = I∞ (wl)−1p

I′∞ (wl)wl

=

(12− 1

p

) ∫RN

(|∇wl |2 + V∞ |wl |2

)dx

=

(12− 1

p

) ∫RN

(|∇wl |2 + V(x) |wl |2 + (V∞ −V(x)) |wl |2

)dx

=

(12− 1

p

) ∫RN

(|∇wl |2 + V(x) |wl |2

)dx + o(1) as l → ∞.

From above, we deduce that

‖u1 + t0wl‖2 = ‖u1‖2 + t20 ‖wl‖2 + o(1)

= ‖u1‖2 +p− 2pc∞

‖wl‖2∣∣∣M2 − ‖u1‖2

∣∣∣+ o(1)

> ‖u1‖2 +∣∣∣M2 − ‖u1‖2

∣∣∣+ o(1) > M2 + o(1)

>

(t−(

u1 + t0wl

‖u1 + t0wl‖

))2

+ o(1) as l → ∞.

Thus t− ((u1 + t0wl) / ‖u1 + t0wl‖) / ‖u1 + t0wl‖ < 1, u1 + t0wl ∈ U2.

Multiple positive solutions for Schrödinger problems 19

We are now in a position to give the proof of Theorem 1.1.

Proof of Theorem 1.1. From Lemma 2.12, we know if | f |q∗ |g|(2−q)/(p−2)∞ ∈ (0, σ∗), then there

is a minimizing sequence un ⊂ N−, which is a (PS)c2 sequence in E. By Lemma 3.5,c2 < c1 + c∞, so from Proposition 3.3, there is a strongly convergent subsequence, still denotedby un, such that un → u2 in E as n → ∞. By Lemma 2.8 the set N− is closed, we knowu2 ∈ N−. Thus, I(u2) = limn→∞ I (un) = infu∈N− I(u). Since I (u2) = I (|u2|) and |u2| ∈ N−,then, without loss of generality, we may assume that u2 is positive. Lemma 2.14 implies thatu2 is a positive solution of Equation (1.1). This with Lemmas 2.7 and 3.4 completes the proofof Theorem 1.1.

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful com-ments that have led to the present improved version. This work is partially supported byNatural Science Foundation of China (Nos: 11371090, 11571140, 11601204, 11671077). Thefirst author is also supported by Natural Science Foundation of Jiangsu Education Committee(No. 18KJB110002).

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